#
Detecting
model categories among Quillen categories using homotopies

##
Seunghun Lee

A model category has two weak factorizations, a pair of cofibrations
and trivial fibrations and a pair of trivial cofibrations and
fibrations. Then the class of weak equivalences is the set W
consisting of the morphisms that can be decomposed into trivial
cofibrations followed by trivial fibrations. One can build a model
category out of such two weak factorizations by defining the class
of weak equivalences by W as long as it satisfies the two out
of three property. In this note we show that given a category with
two weak factorizations, if every object is fibrant and cofibrant,
W satisfies the two out of three property if and only if W
is closed under the homotopies.

Keywords:
Model category, Quillen category, weak equivalence, two out
of three property, homotopy

2020 MSC:
Primary 18N40; Secondary 55U35

*Theory and Applications of Categories,*
Vol. 38, 2022,
No. 1, pp 1-26.

Published 2022-01-06.

http://www.tac.mta.ca/tac/volumes/38/1/38-01.pdf

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